-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5


2. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

3. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 592 549 757 444 051;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 592 549 757 444 051 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 185 099 514 888 102;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 185 099 514 888 102 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 346 370 199 029 776 204;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 346 370 199 029 776 204 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 692 740 398 059 552 408;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 692 740 398 059 552 408 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 385 480 796 119 104 816;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 385 480 796 119 104 816 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 770 961 592 238 209 632;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 770 961 592 238 209 632 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 541 923 184 476 419 264;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 541 923 184 476 419 264 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 083 846 368 952 838 528;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 083 846 368 952 838 528 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 167 692 737 905 677 056;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 167 692 737 905 677 056 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 335 385 475 811 354 112;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 335 385 475 811 354 112 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 670 770 951 622 708 224;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 670 770 951 622 708 224 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 341 541 903 245 416 448;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 341 541 903 245 416 448 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 090 683 083 806 490 832 896;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 090 683 083 806 490 832 896 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 181 366 167 612 981 665 792;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 181 366 167 612 981 665 792 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 362 732 335 225 963 331 584;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 362 732 335 225 963 331 584 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 725 464 670 451 926 663 168;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 725 464 670 451 926 663 168 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 450 929 340 903 853 326 336;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 450 929 340 903 853 326 336 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 901 858 681 807 706 652 672;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 901 858 681 807 706 652 672 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 803 717 363 615 413 305 344;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 803 717 363 615 413 305 344 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 607 434 727 230 826 610 688;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 607 434 727 230 826 610 688 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 214 869 454 461 653 221 376;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 214 869 454 461 653 221 376 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 429 738 908 923 306 442 752;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 429 738 908 923 306 442 752 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 276 859 477 817 846 612 885 504;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 276 859 477 817 846 612 885 504 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 553 718 955 635 693 225 771 008;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 553 718 955 635 693 225 771 008 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 107 437 911 271 386 451 542 016;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 107 437 911 271 386 451 542 016 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 214 875 822 542 772 903 084 032;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 214 875 822 542 772 903 084 032 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 429 751 645 085 545 806 168 064;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 429 751 645 085 545 806 168 064 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 859 503 290 171 091 612 336 128;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 859 503 290 171 091 612 336 128 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 719 006 580 342 183 224 672 256;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 719 006 580 342 183 224 672 256 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 438 013 160 684 366 449 344 512;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 438 013 160 684 366 449 344 512 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 806 876 026 321 368 732 898 689 024;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 806 876 026 321 368 732 898 689 024 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 613 752 052 642 737 465 797 378 048;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 613 752 052 642 737 465 797 378 048 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 227 504 105 285 474 931 594 756 096;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 227 504 105 285 474 931 594 756 096 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 455 008 210 570 949 863 189 512 192;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 455 008 210 570 949 863 189 512 192 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 910 016 421 141 899 726 379 024 384;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 910 016 421 141 899 726 379 024 384 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 820 032 842 283 799 452 758 048 768;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 820 032 842 283 799 452 758 048 768 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 640 065 684 567 598 905 516 097 536;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 640 065 684 567 598 905 516 097 536 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 280 131 369 135 197 811 032 195 072;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 280 131 369 135 197 811 032 195 072 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 560 262 738 270 395 622 064 390 144;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 560 262 738 270 395 622 064 390 144 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 120 525 476 540 791 244 128 780 288;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 120 525 476 540 791 244 128 780 288 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 241 050 953 081 582 488 257 560 576;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 241 050 953 081 582 488 257 560 576 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 482 101 906 163 164 976 515 121 152;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 482 101 906 163 164 976 515 121 152 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 551 941 864 964 203 812 326 329 953 030 242 304;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 551 941 864 964 203 812 326 329 953 030 242 304 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 103 883 729 928 407 624 652 659 906 060 484 608;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 103 883 729 928 407 624 652 659 906 060 484 608 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 207 767 459 856 815 249 305 319 812 120 969 216;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 207 767 459 856 815 249 305 319 812 120 969 216 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 415 534 919 713 630 498 610 639 624 241 938 432;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 415 534 919 713 630 498 610 639 624 241 938 432 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 831 069 839 427 260 997 221 279 248 483 876 864;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 831 069 839 427 260 997 221 279 248 483 876 864 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 662 139 678 854 521 994 442 558 496 967 753 728;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 662 139 678 854 521 994 442 558 496 967 753 728 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 324 279 357 709 043 988 885 116 993 935 507 456;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 324 279 357 709 043 988 885 116 993 935 507 456 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 648 558 715 418 087 977 770 233 987 871 014 912;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 648 558 715 418 087 977 770 233 987 871 014 912 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 297 117 430 836 175 955 540 467 975 742 029 824;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 297 117 430 836 175 955 540 467 975 742 029 824 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 594 234 861 672 351 911 080 935 951 484 059 648;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 594 234 861 672 351 911 080 935 951 484 059 648 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 861 188 469 723 344 703 822 161 871 902 968 119 296;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 861 188 469 723 344 703 822 161 871 902 968 119 296 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 722 376 939 446 689 407 644 323 743 805 936 238 592;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 722 376 939 446 689 407 644 323 743 805 936 238 592 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 444 753 878 893 378 815 288 647 487 611 872 477 184;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 444 753 878 893 378 815 288 647 487 611 872 477 184 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 889 507 757 786 757 630 577 294 975 223 744 954 368;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 889 507 757 786 757 630 577 294 975 223 744 954 368 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 779 015 515 573 515 261 154 589 950 447 489 908 736;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 779 015 515 573 515 261 154 589 950 447 489 908 736 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 558 031 031 147 030 522 309 179 900 894 979 817 472;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 558 031 031 147 030 522 309 179 900 894 979 817 472 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 116 062 062 294 061 044 618 359 801 789 959 634 944;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 116 062 062 294 061 044 618 359 801 789 959 634 944 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 232 124 124 588 122 089 236 719 603 579 919 269 888;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 232 124 124 588 122 089 236 719 603 579 919 269 888 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 464 248 249 176 244 178 473 439 207 159 838 539 776;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 464 248 249 176 244 178 473 439 207 159 838 539 776 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 120 928 496 498 352 488 356 946 878 414 319 677 079 552;
  • 63) 0.083 112 179 287 126 714 097 496 584 120 928 496 498 352 488 356 946 878 414 319 677 079 552 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 241 856 992 996 704 976 713 893 756 828 639 354 159 104;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


5. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 11 positions to the right, so that only one non zero digit remains to the left of it:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) × 20 =

1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010(2) × 2-11


8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 1 (a negative number)

Exponent (unadjusted): -11

Mantissa (not normalized):
1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-11 + 2(11-1) - 1 =

(-11 + 1 023)(10) =

1 012(10)


10. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 012 ÷ 2 = 506 + 0;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1012(10) =

011 1111 0100(2)


12. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010 =

1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1111 0100

Mantissa (52 bits) =
1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


Decimal number -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 296 274 878 722 025 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0100 - 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100